SPECIFIC CONDUCTANCE: how to calculate, to use, and the pitfalls

The specific conductance or electrical conductivity (SC, μS/cm) of a solution is easy to measure and very useful for checking parameters of the aqueous model or defining association constants for new species.
PHREEQC calculates the SC of a solution from the concentration and the diffusion coefficient of the charged species, and applies a correction for ionic strength. Results obtained with the traditional ion-association and the Pitzer model agree very well with measurements of a large variety of solutions, spanning a range of conductances from 10 to 160 000 μS/cm. The figure compares data provided by P.J. Stuyfzand with calculated values.

The standard deviations are 14 and 23 μS/cm with phreeqc.dat and pitzer.dat, respectively, for 185 analyses with SC < 10000 μS/cm.

Generally, the SC's calculated with Pitzer are somewhat higher. With Pitzer's coefficients, the activity of solutes can be calculated well at high salinities, and complexes are usually omitted (although physically, they may be real). On the other hand, for obtaining correct activities at high solute concentrations with the Debye-Hückel method, complexes must often be included. The complexes combine a cation and an anion in a less-charged entity, with the result that the SC decreases. Thus, the stability constants of the complexes can be obtained by fitting the SC's of simple solutions, with the consequence that the SC is better approximated by the Debye-Hückel method. But a caveat is that the molar conductivities of the ions, at concentrations > 1 M, change in a manner that is insufficiently understood. The uncertainty that this brings in the calculation of the complexes permeates in the calculation of the SC of mixtures, and thus of natural waters, especially when the SO42- concentration is high.

The principle of the calculation is that the molar conductivity of a solute species and its diffusion coefficient are related by:

where Λ0m is the molar conductivity (S/m / (mol/m3)), z the charge number (-), F is Faraday's constant (Coulomb/mol), R the gas constant (J/oK/mol), T the absolute temperature (K), and Dw the diffusion coefficient (m2/s). Multiplying the molar conductivity with the concentration m and summing up for all the solutes, gives an estimate of the specific electrical conductance of the solution:
SC = Σ (Λ0m m)
The only problem is that the molar conductivity changes with the concentration.
Kohlrausch's law states that the equivalent conductivity decreases with the square root of the concentration:

Λeq = Λ0eq - Κ (|z| m)0.5
where Κ is Kohlrausch's constant. We could use Λeq (= Λm / |z|) instead of Λ0m to calculate SC. Alternatively, we can stick with Λ0m, but correct the molar concentration with an electrochemical activity coefficient.
Of course, the two methods should produce the same result:
SC = Σ ((Λ0eq - Κ (|z| m)0.5) |z| m) = Σ (Λ0m γsc m)
where γsc is the activity coefficient that corrects the molar concentration or the diffusion coefficient in electro-migration.
Up to version 3.3.9, PHREEQC used to multiply the logarithm of the Debye-Hückel activity coefficient with 0.6 / |z|0.5 for ionic strength I < 0.36 |z|, and I0.5 / |z| otherwise.
Later versions use:
where A and B are the Debye-Hückel parameters, and a1 and a2 are found from measured transport numbers of Cl- and SC's of binary salt solutions (Appelo, 2017, Cem. Concr. Res. 101, 102-113). The new equation is better for saline waters.

PHREEQC's calculations were checked with data from the Handbook of Chemistry and Physics, which lists SC's for salt solutions at 20oC.

PHREEQC prints SC in the output file and the special BASIC variable SC provides it in keywords USER_PUNCH or _GRAPH. Results calculated with PHREEQC input file salt_sc.phr are graphed on the right. Note that the calculations need diffusion coefficients given under SOLUTION_SPECIES in PHREEQC.DAT and PITZER.DAT. The diffusion coefficients are corrected to temperature T (Kelvin) of the solution with:

        (Dw)T = (Dw)298 × exp(d / T - d / 298) × (T / 298) × (η298 / ηT),

where d is a coefficient and η is the viscosity of water.

The coefficients d, a1 and a2 are entered with parameter -dw in SOLUTION_SPECIES.